Standard Normal Probability Function

A Standard Normal Probability Function is a Gaussian function with parameters $\displaystyle{ \mu=0, \sigma=1 }$.

References

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/normal_distribution#Standard_normal_distribution Retrieved:2016-10-4.
• The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ=0 and σ=1, and it is described by this probability density function: : $\displaystyle{ \phi(x) = \frac{e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}}{\sqrt{2\pi}}\, }$ The factor $\displaystyle{ 1/\sqrt{2\pi} }$ in this expression ensures that the total area under the curve $\displaystyle{ \phi(x) }$ is equal to one. [1] The ½ in the exponent ensures that the distribution has unit variance (and therefore also unit standard deviation). This function is symmetric around x=0, where it attains its maximum value $\displaystyle{ 1/\sqrt{2\pi} }$ ; and has inflection points at +1 and −1. Authors may differ also on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance $\displaystyle{ \sigma^2 = \frac{1}{2} }$, that is : $\displaystyle{ \phi(x) = \frac{e^{-x^2}}{\sqrt\pi}\, }$ Stigler goes even further, defining the standard normal with variance $\displaystyle{ \sigma^2 = \frac{1}{(2\pi)} }$ : : $\displaystyle{ \phi(x) = e^{-\pi x^2} }$
1. For the proof see Gaussian integral

2014

• http://math2.org/math/stat/distributions/z-dist.htm
• QUOTE: The z- is a N(0, 1) distribution, given by the equation: $\displaystyle{ f(z) = \frac{1}{2 \pi} e^{\frac{-z^2}{2}} }$

The area within an interval $\displaystyle{ (a,b) = \operatorname{normalcdf}(a,b) = \int_a^b e^2, dz }$ (It is not integrable algebraically.)

The Taylor expansion of the above assists in speeding up the calculation: $\displaystyle{ \operatorname{normalcdf}(-\infty,z) = \frac{1}{2} + \frac{1}{\sqrt{2 \pi}} \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)2^k k!} }$